Multimodal Integration: fMRI, MRI, EEG, and MEG - Advance Image Processing in Magnetic Resonance Imaging - page 226

delta only if and otherwise. Because the beamforming filter for
the i th dipole is defined independently of the other possible dipoles, index i will
be dropped from the derived results for clarity of presentation.
The constrained minimization, solved using Lagrange multipliers, yields

δ ki = 1 ki = 0 F i

F = ( G CG

− 1

)

− 1

G C

− 1

(8.10)

ı X ı ı X

This solution is equivalent to Equation 8.7, when applied to a single dipole with
the regularization term omitted. Source localization is performed using Equation 8.10
to compute the variance of every dipole

q , which, in the case of uncorrelated dipole

moments, is

ν q = tr (( G CG

− 1

)

− 1

) (8.11)

ı X ı

The noise sensitivity of Equation 8.11 can be reduced by using the noise
variance of each dipole as normalizing factor ν ε = tr(( G C 1

−

)

− 1

) . This pro-

ı ε ı

duces the so-called neural activity index

= ν
ν ε

q

z (8.12)

An alternative beamformer, synthetic aperture magnetometry or SAM [42],
is similar to the LCMV if the orientation of the dipole is defined, but it is quite
different in the case of a dipole with an arbitrary orientation. We define a vector
of lead coefficients as a function of the dipole orientation. This returns a
single vector for the orientation

g i () θ

θ of the i th dipole, as opposed to the earlier

formulation in which the L columns of G ı played a similar role. With this new

formulation, we construct the spatial filter

1

f () θ = g ()( θ CC + λ ε )

− 1

(8.13)

i X

g () θ Cg

− 1

() θ

i Xi

which, under standard assumptions, is an optimal linear estimator of the time
course of the i th dipole. The variance of the dipole, accordingly, is also a function
of , specifically To compute the neuronal activity
index, the original SAM formulation uses a slightly different normalization factor
which yields a different result if the noise variance in

θ νθ () = 1 /( () g θ Cg

− 1

()) θ .

q i Xi

νθ () = f () θ Cf () θ , C ε is

ε ε

not equal across the sensors.
The unknown value of is found via a nonlinear optimization of the neuronal
activity index for the dipole:

θ

νϑ
νϑ

()
()

q

θ = arg max

ϑ

ε

Despite the pitfalls of nonlinear optimization, SAM filtering provides a higher
SNR to LCMV by bringing less than half of the noise power into the solution.